Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 2 / b

Table of contents
- Solution b

Solution

Let

ξ

be the Lagrangian displacement. Linearized mass conservation and adiabaticity give

δ ρ = - ξ \cdot \nabla ρ - ρ \nabla \cdot ξ, δ p = - ξ \cdot \nabla p - γ p \nabla \cdot ξ .

The fixed potential has no Eulerian perturbation, so the linearized momentum equation is

ρ \partial_{t}^{2} ξ = - \nabla δ p - δ ρ \nablaΦ.

For the stated spherical harmonic decomposition,

Δ \equiv \nabla \cdot ξ = \frac{1}{r ^{2}} \frac{d ( r ^{2} ξ _{r} )}{d r} - \frac{ℓ ( ℓ + 1 )}{r ^{2}} ξ_{h} .

Taking radial and horizontal components and using

g = Ω^{2} r

gives

- ρ ω^{2} ξ_{r} = - g δ ρ - \frac{d δ p}{d r}, - ρ ω^{2} ξ_{h} = - δ p,

δ ρ = - ξ_{r} \frac{d ρ}{d r} - ρ Δ, δ p = - ξ_{r} \frac{d p}{d r} - γ p Δ .

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version